A Course in Homological Algebra (Graduate Texts in Mathematics) 〈Vol. 4〉 (2ND)

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A Course in Homological Algebra (Graduate Texts in Mathematics) 〈Vol. 4〉 (2ND)

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  • Springer(1997発売)
  • 外貨定価 EUR 71.64
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  • ポイント 208pt
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  • Springer(1997発売)
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  • 製本 Hardcover:ハードカバー版
  • 商品コード 9780387948232

Full Description


Homological algebra has found a large number of applications in many fields ranging from finite and infinite group theory to representation theory, number theory, algebraic topology and sheaf theory. In the new edition of this broad introduction to the field, the authors address a number of select topics and describe their applications, illustrating the range and depth of their developments. A comprehensive set of exercises is included.

Table of Contents

Preface to the Second Edition                      vii
Introduction 1 (9)
I. Modules 10 (30)
1. Modules 11 (5)
2. The Group of Homomorphisms 16 (2)
3. Sums and Products 18 (4)
4. Free and Projective Modules 22 (4)
5. Projective Modules over a Principal 26 (2)
Ideal Domain
6. Dualization, Injective Modules 28 (3)
7. Injective Modules over a Principal 31 (3)
Ideal Domain
8. Cofree Modules 34 (2)
9. Essential Extensions 36 (4)
II. Categories and Functors 40 (44)
1. Categories 40 (4)
2. Functors 44 (4)
3. Duality 48 (2)
4. Natural Transformations 50 (4)
5. Products and Coproducts; Universal 54 (5)
Constructions
6. Universal Constructions (Continued); 59 (4)
Pull-backs and Push-outs
7. Adjoint Functors 63 (6)
8. Adjoint Functors and Universal 69 (5)
Constructions
9. Abelian Categories 74 (7)
10. Projective, Injective, and Free Objects 81 (3)
III. Extensions of Modules 84 (32)
1. Extensions 84 (5)
2. The Functor Ext 89 (5)
3. Ext Using Injectives 94 (3)
4. Computation of some Ext-Groups 97 (2)
5. Two Exact Sequences 99 (7)
6. A Theorem of Stein-Serre for Abelian 106(3)
Groups
7. The Tensor Product 109(3)
8. The Functor Tor 112(4)
IV. Derived Functors 116(50)
1. Complexes 117(4)
2. The Long Exact (Co) Homology Sequence 121(3)
3. Homotopy 124(2)
4. Resolutions 126(4)
5. Derived Functors 130(6)
6. The Two Long Exact Sequences of Derived 136(3)
Functors
7. The Functors Extn Using Projectives 139(4)
8. The Functors Extn Using Injectives 143(5)
9. Extn and n-Extensions 148(8)
10. Another Characterization of Derived 156(4)
Functors
11. The Functor Torn 160(2)
12. Change of Rings 162(4)
V. The Kunneth Formula 166(18)
1. Double Complexes 167(5)
2. The Kunneth Theorem 172(5)
3. The Dual Kunneth Theorem 177(3)
4. Applications of the Kunneth Formulas 180(4)
VI. Cohomology of Groups 184(45)
1. The Group Ring 186(2)
2. Definition of (Co) Homology 188(3)
3. H0, H0 191(1)
4. H1, H1 with Trivial Coefficient Modules 192(2)
5. The Augmentation Ideal, Derivations, 194(3)
and the Semi-Direct Product
6. A Short Exact Sequence 197(3)
7. The (Co) Homology of Finite Cyclic 200(2)
Groups
8. The 5-Term Exact Sequences 202(2)
9. H2, Hopf's Formula, and the Lower 204(2)
Central Series
10. H2 and Extensions 206(4)
11. Relative Projectives and Relative 210(3)
Injectives
12. Reduction Theorems 213(1)
13. Resolutions 214(5)
The (Co) Homology of a Coproduct 219(2)
15. The Universal Coefficient Theorem and 221(2)
the (Co) Homology of a Product
16. Groups and Subgroups 223(6)
VII. Cohomology of Lie Algebras 229(26)
1. Lie Algebras and their Universal 229(5)
Enveloping Algebra
2. Definition of Cohomology; H0, H1 234(3)
3. H2 and Extensions 237(2)
4. A Resolution of the Ground Field K 239(5)
5. Semi-simple Lie Algebras 244(3)
6. The two Whitehead Lemmas 247(4)
7. Appendix: Hilbert's Chain-of-Syzygies 251(4)
Theorem
VIII. Exact Couples and Spectral Sequences 255(51)
1. Exact Couples and Spectral Sequences 256(5)
2. Filtered Differential Objects 261(4)
3. Finite Convergence Conditions for 265(4)
Filtered Chain Complexes
4. The Ladder of an Exact Couple 269(7)
5. Limits 276(5)
6. Rees Systems and Filtered Complexes 281(7)
7. The Limit of a Rees System 288(3)
8. Completions of Filtrations 291(6)
9. The Grothendieck Spectral Sequence 297(9)
IX. Satellites and Homology 306(25)
1. Projective Classes of Epimorphisms 307(2)
2. E-Derived Functors 309(3)
3. E-Satellites 312(6)
4. The Adjoint Theorem and Examples 318(2)
5. Kan Extensions and Homology 320(7)
6. Applications: Homology of Small 327(4)
Categories, Spectral Sequences
X. Some Applications and Recent Developments 331(26)
1. Homological Algebra and Algebraic 331(4)
Topology
2. Nilpotent Groups 335(4)
3. Finiteness Conditions on Groups 339(5)
4. Modular Representation Theory 344(5)
5. Stable and Derived Categories 349(8)
Bibliography 357(2)
Index 359